Quantum information can be negative

Even the most ignorant cannot know less than nothing. After all, negative
information makes no sense. But, although this may be true in the
everyday
world we are accustomed to, negative information
does exist in the quantum world. Small objects such as atoms, molecules
and electrons behave radically different than larger objects -- they obey
the laws of quantum mechanics.

What could negative information possibly mean? In short, after I
send
you negative information, you
will know less.
Such strange situations can occur because what it means to know something
is very different in the quantum world. In the quantum world, we can
know
too much, and it is in these situations where one
finds negative information. Negative
information turns out to be precisely the right amount to cancel the fact
that we know too much.

While all this might appear to be very mysterious
(not to mention, an abuse of the word know!), negative
information, can be put
on a
rigorous footing. I will try to explain how to do so here,
in a manner which I hope is accessible to all. This description is
intended
for those who have an interest in this subject, but may not have a
background in quantum information theory. Most of this text
should be
understandable by anyone willing to put in a bit of effort, and the rest
should be understandable by anyone with some knowledge of quantum
mechanics (or by anyone willing to put in a lot of effort). So if
there are parts
which continue to be unclear after some time, please let me know
[email J.Oppenheim (at)
damtp.cam.ac.uk], and
I can modify this text to make things clearer. An executive summary
of the result can be found at the end of this text here
if you get impatient.

In order to concentrate on the main points, I have sacrificed some
precision, and
even some accuracy, so those with a modest
background in physics should
first try our recent article on negative information, which is
available in Naturehere.
It was written with
Michal
Horodecki, and Andreas
Winter.
Patrick Hayden has written
a commentary on it here,
and a short description of the contents can be found in this
piece by Andreas Trabesinger. The original version of the paper can
be obtained at the pre-print arxiv here. It has a cartoon
and George Orwell quotes, which were deemed inappropriate for Nature.
We will shortly finish a more technical
account which has the full proofs, calculations and details.
Read the rest of this article